Is it permissible to perform the night vigil (Qiyam al-Layl) for someone who has not slept since the previous night?

Every prayer performed by a Muslim at night is considered part of Qiyam al-Layl (night vigil) and is rewarded. As for performing it after sleeping, many scholars call this "Tahajjud," and it is the best and most rewarding type of Qiyam al-Layl. Tahajjud is prayer performed after a period of sleep, as indicated by the hadith of Al-Hajjaj ibn Amr (may Allah be pleased with him).